On the abelian complexity of the Rudin-Shapiro sequence

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function ρ(n), which satisfies certain recurrence relations. As a consequence, the abelian complexity function is 2-regular. Further, we prove that the box dimension of the graph of the asymptotic function λ(x) is 3/2, where λ(x)=limk→∞⁡ρ(4kx)/4kx and ρ(x)=ρ(⌊x⌋) for every x>0.